I'm a nuclear engineer, and "sort of" really is the answer. If you want to know "How?", I've explained it below. If you want to just hear a "Yes/No." then Yes.
I've checked out their diagrams and supplies, they seem legitimate and should actually work. I can't say how well the system would work. The Oxygen Generator systems I worked on used 2000V-3000Volts and up to 900Amps. I am not sure if the 12V in the car would actually produce enough H2 to be useful. But here's the exlpanation. I would actually recommend you try it, with only 12V, there's not going to be enough H2 to destroy your engine and no water will get put into it, so no damage there.
Oxygen generators and Hydrogren generators(Electrolyzer) are the same thing. On the submarine, we had no use for the explosive H2, so we pumped it overboard. But in this case, the H2 would be used and O2 would just be the same as O2 anywhere else.
At the very worst, you would install the kit and not be happy and get a refund. They use PayPal, so it's a legit business and they tell you how to track them to verify their business.
The engines we have are spaced and designed to run on explosive petroleum fuel(gas) or flammable petrol(diesel).
Adding a more explosive compound will reduce the need for the gas.
So YES, it is possible to run it on water or to mix gas and H2 to improve efficiency.
The process is called electrolysis. If you take pure water, which does not conduct electricity, and put a very strong voltage across it(like from the coil in your car that powers the spark plugs) then the Hydrogen and Oxygen separates. The two H2 molecules and single O2 molecule is gas, which comes from two H20 molecules that were seperated.
(2)H20 = (2)H2 + (2)O2
The H2 and O2 are now stable and can be kept separate. If the H2 is allowed to seep into the air intake of a gas engine it would make the fuel more potent and you could produce more power using less gasoline.
That's all great in theory, but in practice, it is a little tougher to achieve than it sounds. It's hard to get de-ionized water, it requires a pricey deionizing filter to make the water pure enough to do this properly. You can do it with simple purified water but that uses quite a bit more electricity with crappy results.
Secondly, those savvy enough to invent their own system have blown their gaskets and the like by putting too much H2 in the engine.
By all means, buy the kit and see if you save money.
I drive an '06 Nissan Sentra and get 25mpg-city/38mpg-Hwy. I am a big skeptic of all these things because of so many scams and such. So if you do install this kit I would appreciate knowing about it and your results. I wouldn't install the kit solely due to the skepticism.
The phrase comes to mind "Never ask an engineer what he does, because he will tell you."